0.1 By Month

0.2 CCAMLR Units

0.3 Distance

0.4 Time

0.5 Velocity

0.6 Angles

1 Correlated random walk

Process Model

\[ d_{t} \sim T*d_{t-1} + Normal(0,\Sigma)\] \[ x_t = x_{t-1} + d_{t} \]

1.1 Parameters

For each individual:

\[\theta = \text{Mean turning angle}\] \[\gamma = \text{Move persistence} \]

For both behaviors process variance is: \[ \sigma_{latitude} = 0.1\] \[ \sigma_{longitude} = 0.1\]

1.2 Behavioral States

\[ \text{For each individual i}\] \[ Behavior_1 = \text{traveling}\] \[ Behavior_2 = \text{foraging}\]

\[ \alpha_{i,1,1} = \text{Probability of remaining traveling when traveling}\] \[\alpha_{i,2,1} = \text{Probability of switching from Foraging to traveling}\]

\[\begin{matrix} \alpha_{i,1,1} & 1-\alpha_{i,1,1} \\ \alpha_{i,2,1} & 1-\alpha_{i,2,1} \\ \end{matrix}\]

With the probability of switching states:

\[logit(\phi_{traveling}) = \alpha_{Behavior_{t-1}}\]

\[\phi_{foraging} = 1 - \phi_{traveling} \]

1.3 Continious tracks

The transmitter will often go dark for 10 to 12 hours, due to weather, right in the middle of an otherwise good track. The model requires regular intervals to estimate the turning angles and temporal autocorrelation. As a track hits one of these walls, call it the end of a track, and begin a new track once the weather improves. We can remove any micro-tracks that are less than three days. Specify a duration, calculate the number of tracks and the number of removed points. Iteratively.

How did the filter change the extent of tracks?

Look at the observations were defined into tracks.

sink(“Bayesian/Multi_RW.jags”) cat(" model{

#Constants
pi <- 3.141592653589

##argos observation error##
argos_prec[1:2,1:2] <- inverse(argos_sigma*argos_cov[,])

#Constructing the covariance matrix
argos_cov[1,1] <- 1
argos_cov[1,2] <- sqrt(argos_alpha) * rho
argos_cov[2,1] <- sqrt(argos_alpha) * rho
argos_cov[2,2] <- argos_alpha

for(i in 1:ind){
for(g in 1:tracks[i]){

## Priors for first true location
#for lat long
y[i,g,1,1:2] ~ dmnorm(argos[i,g,1,1,1:2],argos_prec)

#First movement - random walk.
y[i,g,2,1:2] ~ dmnorm(y[i,g,1,1:2],iSigma)

###First Behavioral State###
state[i,g,1] ~ dcat(lambda[]) ## assign state for first obs

#Process Model for movement
for(t in 2:(steps[i,g]-1)){

#Behavioral State at time T
logit(phi[i,g,t,1]) <- alpha_mu[state[i,g,t-1]] 
phi[i,g,t,2] <- 1-phi[i,g,t,1]
state[i,g,t] ~ dcat(phi[i,g,t,])

#Turning covariate
#Transition Matrix for turning angles
T[i,g,t,1,1] <- cos(theta[state[i,g,t]])
T[i,g,t,1,2] <- (-sin(theta[state[i,g,t]]))
T[i,g,t,2,1] <- sin(theta[state[i,g,t]])
T[i,g,t,2,2] <- cos(theta[state[i,g,t]])

#Correlation in movement change
d[i,g,t,1:2] <- y[i,g,t,] + gamma[state[i,g,t]] * T[i,g,t,,] %*% (y[i,g,t,1:2] - y[i,g,t-1,1:2])

#Gaussian Displacement
y[i,g,t+1,1:2] ~ dmnorm(d[i,g,t,1:2],iSigma)
}

#Final behavior state
logit(phi[i,g,steps[i,g],1]) <- alpha_mu[state[i,g,steps[i,g]-1]] 
phi[i,g,steps[i,g],2] <- 1-phi[i,g,steps[i,g],1]
state[i,g,steps[i,g]] ~ dcat(phi[i,g,steps[i,g],])

##  Measurement equation - irregular observations
# loops over regular time intervals (t)    

for(t in 2:steps[i,g]){

# loops over observed locations within interval t
for(u in 1:idx[i,g,t]){ 
zhat[i,g,t,u,1:2] <- (1-j[i,g,t,u]) * y[i,g,t-1,1:2] + j[i,g,t,u] * y[i,g,t,1:2]

#for each lat and long
#argos error
argos[i,g,t,u,1:2] ~ dmnorm(zhat[i,g,t,u,1:2],argos_prec)
}
}
}
}
###Priors###

#Process Variance
iSigma ~ dwish(R,2)
Sigma <- inverse(iSigma)

##Mean Angle
tmp[1] ~ dbeta(20, 20)
tmp[2] ~ dbeta(10, 10)

# prior for theta in 'traveling state'
theta[1] <- (2 * tmp[1] - 1) * pi

# prior for theta in 'foraging state'    
theta[2] <- (tmp[2] * pi * 2)

##Move persistance
# prior for gamma (autocorrelation parameter)
#from jonsen 2016
gamma[2] ~ dbeta(1,5)       ## gamma for state 2
dev ~ dunif(0.1,1)          ## a random deviate to ensure that gamma[1] > gamma[2]
gamma[1] <- gamma[2] + dev

##Behavioral States

#Hierarchical structure across motnhs
#Intercepts
alpha_mu[1] ~ dnorm(0,0.386)
alpha_mu[2] ~ dnorm(0,0.386)

#Variance
alpha_tau[1] ~ dt(0,1,1)I(0,)
alpha_tau[2] ~ dt(0,1,1)I(0,)

#Probability of behavior switching 
lambda[1] ~ dbeta(1,1)
lambda[2] <- 1 - lambda[1]

##Argos priors##
#longitudinal argos error
argos_sigma ~ dunif(0,10)

#latitidunal argos error
argos_alpha~dunif(0,10)

#correlation in argos error
rho ~ dunif(-1, 1)


}"
,fill=TRUE)

sink()

##    user  system elapsed 
##   1.410   0.056 893.438

1.4 Chains

##                           Type     Size     PrettySize  Rows Columns
## jagM            rjags.parallel 75155096  [1] "71.7 Mb"     6      NA
## mdat                data.frame 16339200  [1] "15.6 Mb" 49859      47
## m                        ggmap 13116480  [1] "12.5 Mb"  1280    1280
## b       SpatialPointsDataFrame  4875992   [1] "4.7 Mb"  3354      47
## fccamlr             data.frame  1649608   [1] "1.6 Mb" 41160       7
## sxy                       list  1184688   [1] "1.1 Mb"    32      NA
## mxy                 grouped_df   954216 [1] "931.9 Kb"  2924      52
## d       SpatialPointsDataFrame   939008   [1] "917 Kb"  3354      47
## oxy                 data.frame   882256 [1] "861.6 Kb"  3354      47
## data                      list   724544 [1] "707.6 Kb"     9      NA
##            used  (Mb) gc trigger  (Mb) max used  (Mb)
## Ncells  1552602  83.0    2637877 140.9  2637877 140.9
## Vcells 15509395 118.4   29035403 221.6 35733381 272.7

Look at the convergence of phi, just for an example

Same section of chain for state

Overall relationship between phi and state, nice test of convergence.

1.4.1 Compare to priors

1.5 Parameter Summary

##   parameter         par        mean        lower     upper
## 1  alpha_mu alpha_mu[1] -0.42707156 -2.937763398 1.5979076
## 2  alpha_mu alpha_mu[2]  2.04043806  0.432851690 3.6966692
## 3     gamma    gamma[1]  0.41540786  0.281960880 0.5569139
## 4     gamma    gamma[2]  0.09060005  0.005951651 0.2378229
## 5     theta    theta[1] -0.06096017 -0.257053471 0.1281514
## 6     theta    theta[2]  2.75650002  1.773922586 3.8819279

2 Behavioral Prediction

Relationship between phi and state

2.1 Spatial Prediction

2.2 Confidence in state estimate

Red is traveling, blue is foraging, midpoint is 0

2.3 By individual

2.4 Compared to CMLRR regions

2.5 Autocorrelation in behavior

2.6 Location of Behavior

3 Overlap with Krill Fishery

4 Time spent in grid cell

4.1 Traveling

##                             Type     Size     PrettySize    Rows Columns
## pc                        tbl_df 74459632    [1] "71 Mb" 1430000      10
## a                         tbl_df 19846152  [1] "18.9 Mb"  496000       7
## mdat                  data.frame 16339200  [1] "15.6 Mb"   49859      47
## temp                       ggmap 13116816  [1] "12.5 Mb"    1280    1280
## b         SpatialPointsDataFrame  4875992   [1] "4.7 Mb"    3354      47
## fccamlr               data.frame  1649608   [1] "1.6 Mb"   41160       7
## d         SpatialPointsDataFrame   939008   [1] "917 Kb"    3354      47
## mxy                   data.frame   929032 [1] "907.3 Kb"    2748      58
## oxy                   data.frame   882256 [1] "861.6 Kb"    3354      47
## traveling             data.frame   833808 [1] "814.3 Kb"    2461      58
##            used  (Mb) gc trigger  (Mb) max used  (Mb)
## Ncells  1615418  86.3    2637877 140.9  2637877 140.9
## Vcells 18313403 139.8   41986979 320.4 41979005 320.3